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        <h2 id="食品配方问题"><a href="#食品配方问题" class="headerlink" title="食品配方问题"></a>食品配方问题</h2><p>甲乙丙丁四种原料营养比如下</p>
<div class="table-container">
<table>
<thead>
<tr>
<th></th>
<th style="text-align:center">甲</th>
<th style="text-align:center">乙</th>
<th style="text-align:center">丙</th>
<th style="text-align:center">丁</th>
</tr>
</thead>
<tbody>
<tr>
<td>蛋白质%</td>
<td style="text-align:center">20</td>
<td style="text-align:center">16</td>
<td style="text-align:center">10</td>
<td style="text-align:center">15</td>
</tr>
<tr>
<td>脂肪%</td>
<td style="text-align:center">3</td>
<td style="text-align:center">8</td>
<td style="text-align:center">2</td>
<td style="text-align:center">5</td>
</tr>
<tr>
<td>碳水化合物%</td>
<td style="text-align:center">10</td>
<td style="text-align:center">25</td>
<td style="text-align:center">20</td>
<td style="text-align:center">5</td>
</tr>
</tbody>
</table>
</div>
<p> 做出满足 蛋白质 15%， 脂肪 5%， 碳水化合物 12% 的食品， 求 上述四种原料配比 。</p>
<h2 id="解法"><a href="#解法" class="headerlink" title="解法"></a>解法</h2><h3 id="建立数学模型"><a href="#建立数学模型" class="headerlink" title="建立数学模型"></a>建立数学模型</h3><p>设四种原料占该食物的百分比分别为 $x_1$, $x_2$, $x_3$, $x_4$. 则:</p>
<script type="math/tex; mode=display">
\begin{Bmatrix}
 x_1+&  x_2+&  x_3+&  x_4=&  1  &  （总含量）&\\\\ 
 20x_1 +& 16x_2 + & 10x_3 + &  15x_4 =  & 15  & (蛋白质)&\\\\ 
 3x_1 + & 8x_2 + & 2x_3 + & 5x_4 =  &  5  & (脂肪)&\\\\ 
 10x_1 +&  25x_2 + &   20x_3 + & 20x_4 = &  12  & (碳水化合物)&
\end{Bmatrix}.</script><h2 id="线性方程组解的类型"><a href="#线性方程组解的类型" class="headerlink" title="线性方程组解的类型"></a>线性方程组解的类型</h2><pre class="mermaid">graph LR;
  A(线性方程组解的类型) --> B{是否有解} 
    B --> D(有唯一解)
    B --> E(有多个解)
    B --> F(无解)
    F --> G(找出近似解)

    D --> H{是否合理}
    E-->H
    H-->I(解集的性质)</pre>



<h2 id="线性方程组的类型"><a href="#线性方程组的类型" class="headerlink" title="线性方程组的类型"></a>线性方程组的类型</h2><pre class="mermaid">graph LR;
  A(线性方程组类型) --> B(适定方程组)
  B-->G(唯一解)
  A-->C(欠定方程组)
  C-->E(解不唯一)
  A-->D(超定方程组)
  D-->F(不存在精确解, 可求出近似解)</pre>
      
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              <ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#食品配方问题"><span class="toc-number">1.</span> <span class="toc-text">食品配方问题</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#解法"><span class="toc-number">2.</span> <span class="toc-text">解法</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#建立数学模型"><span class="toc-number">2.1.</span> <span class="toc-text">建立数学模型</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#线性方程组解的类型"><span class="toc-number">3.</span> <span class="toc-text">线性方程组解的类型</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#线性方程组的类型"><span class="toc-number">4.</span> <span class="toc-text">线性方程组的类型</span></a></li></ol>
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